An elementary solution of Gessel's walks in the quadrant
Mireille Bousquet-Mélou
TL;DR
This work provides the first elementary, constructive proof that Gessel's quadrant walks, with steps $\ ightarrow, \\nearrow, \\leftarrow, \\swarrow$, have an algebraic generating function $Q(x,y;t)$ of degree $72$ over $\mathbb{Q}(x,y,t)$, with the specialisation $Q(0,0;t)$ of degree $8$ admitting an explicit parametrization. The method uses a kernel-based functional equation, symmetry of the kernel roots $Y_0,Y_1$, and a generalized quadratic approach to derive an ordinary discrete differential equation for $R(x)=t(Q(x,0)-Q(0,0))$, leading to algebraicity without heavy computer algebra or analytic machinery. A genus-0 parametrization in terms of a variable $T$ (and $Z=\,sqrt{T}$) yields explicit expressions for $Q(0,0)$, $Q(x,0)$, and $Q(0,y)$, enabling an explicit description of the trivariate series $Q(x,y;t)$. The technique extends to other algebraic quadrant models, offering a unified, human-accessible route to algebraicity for several small-step walks and clarifying the algebraic structure underlying quadrant enumeration.
Abstract
Around 2000, Ira Gessel conjectured that the number of lattice walks in the quadrant N^2, starting and ending at the origin (0,0) and taking their steps in {E,NE,W,SW} had a simple hypergeometric form. In the following decade, this problem was recast in the systematic study of walks with small steps (that is,steps in {-1,0,1}^2) confined to the quadrant. The generating functions of such walks are archetypal solutions of partial discrete differential equations.A complete classification of quadrant walks according to the nature of their generating function(algebraic, D-finite or not) is now available, but Gessel'swalks remained mysterious because they were the only model among the 23D-finite ones that had not been given an elementarysolution. Instead, Gessel's conjecture was first proved usingan inventive computer algebra approach in 2008. A year later, the associated three-variate generating function was proved to be algebraic by a computer algebra tour de force. This was re-proved recently using elaborate complex analysis machinery. We give here an elementary and constructive proof. Our approach also solves other quadrant models (with multiple steps) recently proved to be algebraic via computer algebra.